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# On the generalized pythagorean parameters and the applications in Banach spaces

• Let X be a Normed space and $S(X) = \{x \in X : \|\|x\|\| = 1\}$ be the unit sphere of X. Following the previous results for the Pythagorean approach in Banach spaces [5], [6], the generalized parameters $E_{\xi, \eta}(X)=$sup${\alpha_{\eta}(\xi x): x in S(X)\}$, $e_{\xi, \eta}(X)=$inf$\{\alpha_{\eta}(\xi x): x \in S(X)\}$, $F_{\xi, \eta}(X)=$sup${\beta_{\eta}(\xi x): x \in S(X)\}$, and $f_{\xi, \eta}(X)=$inf${\beta_{\eta}(\xi x): x \in S(X)\}$, where $\alpha_{\eta}(\xi x) =$sup${||\xi x + \eta y ||^2+ ||\xi x - \eta y ||^{2}: y \in S(X)\}$, $\beta_{\eta}(\xi x) =$inf${\|\|\xi x + \eta y ||^{2}+ ||\xi x - \eta y ||^{2}: y \in S(X)\}$ and $\xi, \eta > 0$ are defined and studied. The values of these parameters of some classical normed spaces are estimated and the relationship of these parameters with other geometric properties are investigated, and some existing results are extended also.
Mathematics Subject Classification: Primary: 46C05, 52A21 and 46B20.

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